Question

Найдите первый член и знаменатель геометрической прогрессии ( a n ​ ) , в которой a 4 ​ − a 2 ​ = 3 6 , a 5 ​ − a 3 ​ = 7 2 . В ответе запишите шестой член этой прогрессии.

Answer 1

$\displaystyle\bf\\a_{4} -a_{2} =36\\\\a_{5} -a_{3}=72\\\\\\\left \{ {{a_{1}\cdot q^{3} -a_{1}\cdot q=36} \atop {a_{1}\cdot q^{4} -a_{1}\cdotq^{2} =72 }} \right.\\\\\\:\left \{ {{a_{1} q^{2} \cdot (q^{2} -1)=72} \atop {a_{1} q \cdot (q^{2} -1)=36}} \right. \\------------\\q=2\\\\\\a_{1}=\frac{36}{q\cdot(q^{2}-1)} } =\frac{36}{2\cdot (2^{2}-1) } =\frac{18}{3} =6\\\\a_{6} =a_{1} \cdot q^{5} =6\cdot 2^{5} =6\cdot 32=192\\\\Otvet:a_{6}=192$

Answer 2

Finding the First Term and Common Ratio of a Geometric Progression

To find the first term and common ratio of a geometric progression, we can use the given equations:

a4 - a2 = 36 (Equation 1) a5 - a3 = 72 (Equation 2)

Let's solve these equations to find the values of a and n.

Solving the Equations

To find the first term (a) and common ratio (n) of the geometric progression, we can use the following formulas:

aₙ = a₁ * n^(n-1) aₙ₊₁ = a₁ * n^n

Substituting the given values into the formulas, we have:

a₄ - a₂ = a₁ * 4^3 - a₁ * 2^1 = 36 (Equation 1) a₅ - a₃ = a₁ * 5^4 - a₁ * 3^2 = 72 (Equation 2)

Simplifying the equations, we get:

8a₁ - 2a₁ = 36 16a₁ - 9a₁ = 72

Combining like terms, we have:

6a₁ = 36 7a₁ = 72

Solving for a₁, we find:

a₁ = 6

Now, let's find the common ratio (n) using the formula:

n = (aₙ₊₁ / aₙ)^(1/n)

Substituting the values of aₙ₊₁ and aₙ, we have:

n = (a₁ * n^n) / (a₁ * n^(n-1))

Simplifying the equation, we get:

n = n / (n - 1)

Solving for n, we find:

n = 2

Finding the Sixth Term

Now that we have the first term (a₁ = 6) and the common ratio (n = 2), we can find the sixth term (a₆) of the geometric progression using the formula:

aₙ = a₁ * n^(n-1)

Substituting the values, we have:

a₆ = 6 * 2^(6-1) a₆ = 6 * 2^5 a₆ = 6 * 32 a₆ = 192

Therefore, the sixth term of the geometric progression is 192.

Please let me know if there is anything else I can help you with!